The critique of weak negation in constructive mathematics \((\mathrm {CM})\)  goes back to Griss [1]}.
Despite the use of weak negation in Brouwer's intuitionistic mathematics \((\mathrm {INT})\)  and Bishop's constructive
mathematics \((\mathrm {BISH})\) , both, Brouwer and Bishop, developed a positive and strong approach
to many classically negatively defined concepts. An apartness relation \(\ne _X\)  on a set \((X, =_X)\)  is the strong
counterpart to the weak negation \(\lnot (x =_X x{^{\prime }})\)  of its equality, the strong complement \(A^{\ne } := \lbrace x \in X \mid \forall _{a \in A}(x \ne _X a)\rbrace \)  of a subset \(A\)  of \(X\)  is the positive version of its standard,
weak complement \(A^c := \lbrace x \in X \mid \forall _{a \in A}(\lnot (x \in A))\rbrace \) ,
inhabited sets are the strong, non-empty sets, and so on. Recently, Shulman [2]} showed
that most of these strong concepts of \(\mathrm {BISH}\)  “arise automatically from an “antithesis” translation
of affine logic into intuitionistic logic via a “Chu/Dialectica construction”.
Motivated by this work of Shulman, and in relation to a reconstruction of the theory of sets underlying
\(\mathrm {BISH}\)  (see [3]}, [4]}, [5]} and [6]}),
the second author develops in [7]} a treatment of strong negation within
\(\mathrm {BISH}\) , arriving in a heuristic method for the definition of these strong concepts
of \(\mathrm {BISH}\)  similar to Shulman's. For that, the equivalences occurring in the axiom schemata related to strong negation in
\(\mathrm {CLSN}\)  become the definitional clauses of the recursive definition of \(\mathbin {\sim } A\) , with respect to the
inductive definition of formulas \(A\)  in \(\mathrm {BISH}\) .
